Question

these solids are similar. find the volume of the larger solid. h = 4 m v = 56 m³ h = 10 m v =? m³

Explanation:

1. Recall the ratio - volume relationship for similar solids:

• For two similar solids, if the ratio of their corresponding linear dimensions (such as heights) is \(a:b\), the ratio of their volumes is \(a^{3}:b^{3}\).
• Let the height of the smaller solid be \(h_1 = 4m\) and its volume be \(V_1=56m^{3}\), and the height of the larger solid be \(h_2 = 10m\).
• The ratio of the heights of the two similar solids is \(\frac{h_2}{h_1}=\frac{10}{4}=\frac{5}{2}\).
• The ratio of the volumes of the two similar solids is \((\frac{h_2}{h_1})^{3}\). Let the volume of the larger solid be \(V_2\). Then \(\frac{V_2}{V_1}=(\frac{h_2}{h_1})^{3}\).

1. Substitute the known values into the volume - ratio formula:

• We know that \(\frac{V_2}{56}=(\frac{10}{4})^{3}=(\frac{5}{2})^{3}=\frac{125}{8}\).
• To find \(V_2\), we can solve the equation \(V_2 = 56\times\frac{125}{8}\).
• First, simplify \(56\div8 = 7\).
• Then, \(V_2=7\times125 = 875m^{3}\).

Answer:

\(875m^{3}\)